For a one-dimensional flow situation involving a source, a simple closed form solution for the mass density is obtained for the mass continuity equation by introducing a change in variable. Instead of functionalizing the flow in terms of the velocity, the starting time or transit time is used. This type of functionalization allows the individual trajectories to be obtained by algebraic or geometric means, avoiding the nonclosed form solution associated with integrating the velocity. For a time varying system, the integrals of mass, momentum, and energy all involve the mass density expression. By eliminating the mass density expression from these integrals and expressing the velocity in terms of one of the new time parameters, a set of equations is obtained which may be easier to solve than the original set of equations.